It depends on your local jurisdiction and what you are calling a “bathroom remodel”. Some codes specify that if you are replacing more than 33% of the plumbing in a residence then a permit is required. Typically, if you are only replacing surfaces and no alteration to the electric or plumbing is occurring then a permit is not required.
Yes you do!! The city makes sure that not only the Contracter is licensed but installs it correctly.
The answer varies by state , however, if you plan do do any major plumbing, electrical, or structural alterations to your existing room, then you ‘should’ secure the required permits. This will help ensure the work is performed according to current code and will likely prevent under-qualified remodeling professionals from getting in over their heads on a project that they shouldn’t have bid on.